We see that our range gets worse as \(H\)  and \(y\)  increase. This is to be expected. Larger \(H\)  means we have less averaging present in our variance, where we think of \([0,X]\)  as a union of \(X/H\)  disjoint intervals of size \(H\) . This is similar to how the results in [1]}, [2]}, [3]} on squarefrees and on primes hold only for \(H\)  sufficiently small. Larger \(y\)  means \(\alpha _y\)  becomes closer to the indicator of primes, and so we expect our range to deteriorate as \(y\)  grows.
